1. Field of the Invention
The present invention relates to modular polynomial divisions.
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2. Background Art
Computer systems are useful for performing mathematical operations (add, subtract, multiply, divide) on operands. Often the operands are polynomials. A polynomial is a mathematical expression of one or more algebraic terms each of which consists of a constant multiplied by one or more variables raised to a nonnegative integral power (e.g. a+bx+cx2). The task of performing mathematical operations on polynomial operands is difficult in the sense that it is not simply a matter of multiplying or dividing two simple numbers. There are a number of schemes that provide methods for performing mathematical operations on polynomials. However, there are situations for which no suitable schemes have been provided.
One situation that requires the manipulation of polynomials is the encryption and decryption of data in a cryptosystem and digital signatures for verification of the sender. A cryptosystem is a system for sending a message from a sender to a receiver over a medium so that the message is xe2x80x9csecurexe2x80x9d, that is, so that only the intended receiver can recover the message. A cryptosystem converts a message, referred to as xe2x80x9cplaintextxe2x80x9d into an encrypted format, known as xe2x80x9cciphertext.xe2x80x9d The encryption is accomplished by manipulating or transforming the message using a xe2x80x9ccipher keyxe2x80x9d or keys. The receiver xe2x80x9cdecryptsxe2x80x9d the message, that is, converts it from ciphertext to plaintext, by reversing the manipulation or transformation process using the cipher key or keys. So long as only the sender and receiver have knowledge of the cipher key, such an encrypted transmission is secure.
A digital signature is a bit-stream generated by a cryptosystem. It is attached to a message such that a receiver of the message can verify with the bit-stream and be assured that the message was indeed originated from the sender it claims to be. A xe2x80x9cclassicalxe2x80x9d cryptosystem is a cryptosystem in which the enciphering information can be used to determine the deciphering information. To provide security, a classical cryptosystem requires that the enciphering key be kept secret and provided to users of the system over secure channels. Secure channels, such as secret couriers, secure telephone transmission lines, or the like, are often impractical and expensive.
A system that eliminates the difficulties of exchanging a secure enciphering key is known as xe2x80x9cpublic key encryption.xe2x80x9d By definition, a public key cryptosystem has the property that someone who knows only how to encipher a message cannot use the enciphering key to find the deciphering key without a prohibitively lengthy computation. An enciphering function is chosen so that once an enciphering key is known, the enciphering function is relatively easy to compute. However, the inverse of the encrypting transformation function is difficult, or computationally infeasible, to compute. Such a function is referred to as a xe2x80x9cone way functionxe2x80x9d or as a xe2x80x9ctrap door function.xe2x80x9d In a public key cryptosystem, certain information relating to the keys is public. This information can be, and often is, published or transmitted in a non-secure manner. Also, certain information relating to the keys is private. This information may be distributed over a secure channel to protect its privacy, (or may be created by a local user to ensure privacy). Some of the cryptosystems that have been developed include the RSA system, the Massey-Omura system, and the El Gamal system.
Elliptic Curves
Another form of public key cryptosystem is referred to as an xe2x80x9celliptic curvexe2x80x9d cryptosystem. An elliptic curve cryptosystem is based on points on an elliptic curve E defined over a finite field F. Elliptic curve cryptosystems rely for security on the difficulty in solving the discrete logarithm problem. An advantage of an elliptic curve cryptosystem is there is more flexibility in choosing an elliptic curve than in choosing a finite field. Nevertheless, elliptic curve cryptosystems have not been widely used in computer-based public key exchange systems due to their late discovery and the mathematical complexity involved. Elliptic curve cryptosystems are described in xe2x80x9cA Course in Number Theory and Cryptographyxe2x80x9d (Koblitz, 1987, Springer-Verlag, N.Y.).
In practice an Elliptic Curve group over Fields F(2m) is formed by choosing a pair of a and b coefficients, which are elements within F(2m). The group consists of a finite set of points P(x,y) which satisfy the elliptic curve equation
y2+xy=x3+ax2+b
together with a point at infinity, O. The coordinates of the point, x andy, are elements of F(2m) represented in m-bit strings. Since F(2m) operates on bit strings and the field has a characteristic 2, computers can perform arithmetic in this field very efficiently. The arithmetic in F(2m) can be defined in either a standard basis representation or optimal normal basis representation. This description uses the standard basis representations for purposes of discussion. All elliptic curve point coordinates are represented as polynomials with binary coefficients.
The Elliptic Curve Cryptosystem relies upon the difficulty of the Elliptic Curve Discrete Logarithm Problem (ECDLP) to provide its effectiveness as a cryptosystem. Using multiplicative notation, the problem can be described as: given points P and Q in the group, find a number k such that PK=Q; where k is called the discrete logarithm of Q to the base P. Using additive notation, the problem becomes: given two points P and Q in the group, find a number k such that kP=Q.
In an Elliptic Curve Cryptosystem, the large integer k is kept private and is often referred to as the secret key. The point Q together with the base point P are made public and are referred to as the public key. The security of the system, thus, relies upon the difficulty of deriving the secret k, knowing the public points P and Q. The main factor that determines the security strength of such a system is the size of its underlying finite field. In a real cryptographic application, the underlying field is made so large that it is computationally infeasible to determine k in a straight forward way by computing all the multiples of P until Q is found.
The core of the elliptic curve geometric arithmetic is an operation called scalar multiplication which computes kP by adding together k copies of the point P. The scalar multiplication is performed through a combination of point-doubling and point-addition operations. The point-addition operation adds two distinct points together and the point-doubling operation adds two copies of a point together. To compute, for example, 11P=(2*(2*(2P)))+2P=P, it would take 3 point-doublings and 2 point-additions.
Point-doubling and point-addition calculations require special operations when dealing with polynomial operands. Algebraic schemes for accomplishing these operations are illustrated below in Table 1.
The two equations for S in the table are called the slope-equations. Computation of a slope equation requires one modular polynomial inversion (1/X mod M) where M is an irreducible polynomial and one modular polynomial multiplication (*Y mod M). Because the operands are polynomials, these operations are typically done back-to-back as two separate operations. There exist algorithms and solutions to calculate the modular inversion 1/X mod M and the modular multiplication *Y mod M. After the result of the modular inversion is calculated, the multiplication *Y mod M is performed. Of course, algebraically (1/X*Y) mod M is the same as Y/X mod M. However, there is currently no technique for calculating modular Y/X in one operation when the operands are polynomial functions. These two field operations, the inversion and the multiply, are expensive computationally because they require extensive CPU cycles for the manipulation of two large polynomials modular a large irreducible polynomial. Today, it is commonly accepted that a point-doubling and point-addition operation each requires one inversion, two multiplies, a square, and several additions. To date there are techniques to compute modular inversions, and techniques to trade expensive inversions for multiplies by performing the operations in projective coordinates. There have been no efficient hardware oriented techniques suggested to compute a modular division directly which can be used to perform point doubling and point addition operations.
The present invention provides a method for performing an inversion and multiply in a single operation as a polynomial divide operation. As a result, the invention reduces the number of mathematical operations needed to perform point doubling and point addition operations. An elliptic curve cryptosystem using the present invention can be made to operate more efficiently using the present invention. An elliptic curve crypto-accelerator can be implemented using the present invention to dramatically enhance the performance of the elliptic curve cryptosystem.
The invention uses five registers A, B, U, V, and M, to accomplish a polynomial divide operation. Four registers A, B, U, and V are initialized with values so that the registers maintain a number of invariant relationships. The registers store initial values a(t)=x(t), u(t)=y(t), b(t)=prime(t), and v(t)=0. Here the polynomials in registers A, U, B, and V are denoted as a(t), u(t), b(t), and v(t), respectively. Register M stores the irreducible polynomial prime(t). By applying a series of invariant operations to the registers, the register values are systematically reduced until registers A and B have a value of one. At that point, register U stores a value which represents y(t)/x(t) mod prime(t), solving the polynomial division.